Newform orbit 288.3.bd.a

• Label: 288.3.bd.a
• Level: $288$
• Weight: $3$
• Character orbit: 288.bd
• Analytic conductor: $7.847$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	288.bd (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.84743161358\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(94\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 4 q^{2} - 8 q^{3} - 4 q^{4} - 4 q^{5} - 8 q^{6} - 4 q^{7} - 16 q^{8} - 8 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−2.00000	+	0.00427522i	2.00426	+	2.23225i	3.99996	−	0.0171008i	−2.40810	+	0.317032i	−4.01805	−	4.45592i	−3.07666	−	11.4823i	−7.99984	+	0.0513023i	−0.965891	+	8.94802i	4.81483	−	0.644358i
43.2	−1.99885	−	0.0677146i	−2.73571	+	1.23121i	3.99083	+	0.270703i	6.64775	−	0.875193i	5.55166	−	2.27577i	−2.37443	−	8.86148i	−7.95875	−	0.811334i	5.96823	−	6.73648i	−13.3471	+	1.29923i
43.3	−1.99272	+	0.170448i	2.91417	−	0.712475i	3.94190	−	0.679310i	−2.73993	+	0.360719i	−5.68569	+	1.91648i	−0.488943	−	1.82476i	−7.73932	+	2.02556i	7.98476	−	4.15255i	5.39844	−	1.18583i
43.4	−1.99237	+	0.174547i	−0.724495	−	2.91120i	3.93907	−	0.695523i	2.32576	−	0.306192i	1.95160	+	5.67373i	1.90380	+	7.10508i	−7.72667	+	2.07329i	−7.95021	+	4.21831i	−4.58033	+	1.01600i
43.5	−1.98779	−	0.220648i	0.943533	−	2.84776i	3.90263	+	0.877203i	−6.73753	+	0.887012i	−2.50390	+	5.45257i	1.41631	+	5.28573i	−7.56406	−	2.60480i	−7.21949	−	5.37391i	13.5885	−	0.276576i
43.6	−1.96297	−	0.383066i	1.94050	−	2.28789i	3.70652	+	1.50390i	8.45707	−	1.11339i	−4.68557	+	3.74773i	−1.08242	−	4.03965i	−6.69970	−	4.37195i	−1.46890	−	8.87932i	−17.0275	−	1.05406i
43.7	−1.96231	+	0.386456i	−0.0727852	+	2.99912i	3.70130	−	1.51669i	5.52539	−	0.727431i	−1.01620	−	5.91332i	1.36798	+	5.10538i	−6.67696	+	4.40660i	−8.98940	−	0.436583i	−10.5614	+	3.56276i
43.8	−1.94873	−	0.449946i	−2.99953	−	0.0530887i	3.59510	+	1.75365i	−1.59655	+	0.210190i	5.82139	+	1.45308i	2.04120	+	7.61785i	−6.21683	−	5.03498i	8.99436	+	0.318482i	3.20583	+	0.308759i
43.9	−1.91416	−	0.579653i	−2.11566	−	2.12696i	3.32800	+	2.21910i	−8.26835	+	1.08855i	2.81681	+	5.29770i	−3.01589	−	11.2555i	−5.08402	−	6.17679i	−0.0479595	+	8.99987i	16.4579	+	2.70912i
43.10	−1.90743	+	0.601414i	1.89207	+	2.32810i	3.27660	−	2.29431i	−6.58740	+	0.867248i	−5.00916	−	3.30278i	2.63503	+	9.83407i	−4.87007	+	6.34684i	−1.84011	+	8.80988i	12.0435	−	5.61597i
43.11	−1.85750	+	0.741420i	−2.33943	+	1.87805i	2.90059	−	2.75437i	−5.50595	+	0.724873i	2.95306	−	5.22297i	−0.925207	−	3.45292i	−3.34570	+	7.26680i	1.94585	−	8.78713i	9.68986	−	5.42867i
43.12	−1.81320	−	0.843981i	−0.764114	+	2.90106i	2.57539	+	3.06061i	−8.52605	+	1.12248i	3.83393	−	4.61530i	1.33770	+	4.99237i	−2.08660	−	7.72309i	−7.83226	−	4.43348i	16.4068	+	5.16055i
43.13	−1.81061	+	0.849515i	−2.43710	−	1.74944i	2.55665	−	3.07629i	−1.50653	+	0.198339i	5.89883	+	1.09720i	−0.597966	−	2.23164i	−2.01575	+	7.74188i	2.87895	+	8.52711i	2.55926	−	1.63894i
43.14	−1.80928	−	0.852346i	2.66291	+	1.38162i	2.54701	+	3.08427i	3.27943	−	0.431745i	−3.64034	−	4.76948i	1.86766	+	6.97022i	−1.97940	−	7.75126i	5.18223	+	7.35830i	−6.30141	−	2.01406i
43.15	−1.79999	−	0.871791i	0.0129107	+	2.99997i	2.47996	+	3.13844i	2.27010	−	0.298865i	2.59211	−	5.41119i	−1.05461	−	3.93587i	−1.72786	−	7.81118i	−8.99967	+	0.0774636i	−4.34672	−	1.44110i
43.16	−1.76692	+	0.937021i	2.89256	−	0.795660i	2.24398	−	3.31127i	5.32494	−	0.701042i	−4.36537	+	4.11626i	1.59802	+	5.96391i	−0.862196	+	7.95340i	7.73385	−	4.60300i	−8.75184	+	6.22827i
43.17	−1.69372	−	1.06364i	−2.17254	−	2.06883i	1.73736	+	3.60300i	4.67735	−	0.615785i	1.47919	+	5.81481i	−0.799793	−	2.98487i	0.889673	−	7.95038i	0.439869	+	8.98924i	−8.57709	−	3.93203i
43.18	−1.67678	+	1.09014i	−2.90163	−	0.761951i	1.62318	−	3.65586i	7.60746	−	1.00154i	5.69602	−	1.88556i	1.35645	+	5.06234i	1.26369	+	7.89956i	7.83886	+	4.42179i	−11.6642	+	9.97257i
43.19	−1.65998	+	1.11556i	0.669363	−	2.92437i	1.51106	−	3.70360i	−0.912859	+	0.120180i	2.15118	+	5.60111i	−2.26113	−	8.43865i	1.62325	+	7.83359i	−8.10391	−	3.91493i	1.38126	−	1.21784i
43.20	−1.54979	+	1.26418i	2.55220	+	1.57680i	0.803679	−	3.91843i	6.06580	−	0.798577i	−5.94872	+	0.782742i	−2.92729	−	10.9248i	3.70809	+	7.08873i	4.02741	+	8.04860i	−8.39114	+	8.90590i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
32.h	odd	8	1	inner
288.bd	odd	24	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	288.3.bd.a	✓	752
9.c	even	3	1	inner	288.3.bd.a	✓	752
32.h	odd	8	1	inner	288.3.bd.a	✓	752
288.bd	odd	24	1	inner	288.3.bd.a	✓	752

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
288.3.bd.a	✓	752	1.a	even	1	1	trivial
288.3.bd.a	✓	752	9.c	even	3	1	inner
288.3.bd.a	✓	752	32.h	odd	8	1	inner
288.3.bd.a	✓	752	288.bd	odd	24	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(288, [\chi])\).